PACKET Unit 4 Honors ICM Functions and Limits 1

Transcription

1 PACKET Unit 4 Honors ICM Functions and Limits 1 Day 1 Homework For each of the rational functions find: a. domain b. -intercept(s) c. y-intercept Graph #8 and #10 with at least 5 EXACT points. 1. f 6. f 1. h 9 -intercept(s): -intercept(s): -intercept(s): 4. g f f 1 6 -intercept(s): -intercept(s): -intercept(s): 7. f f 1 Graph #8 -intercept(s): -intercept(s): 9. f 9 f Graph #10 -intercept(s): -intercept(s):

2 PACKET Unit 4 Honors ICM Functions and Limits Homework Day In Eercises 9-, graph each function including and y intercepts and give the domain. If the function is discontinuous, state the discontinuity and tell whether it is removable or nonremovable. 9. -int: f ( ) 4 y-int: f( ) ( )( 1) -int: y-int: Domain: Domain: g ( ) int: y-int: 4 h ( ) ( 1)( 1) 15. -int: y-int: Domain: Domain:

3 PACKET Unit 4 Honors ICM Functions and Limits 16. f ( ) int: y-int: Domain: 1. g ( ) -int: y-int:. f( ) -int: y-int: Domain: Domain:

4 PACKET Unit 4 Honors ICM Functions and Limits 4 Asymptote Lab Classwork Day Eamine and write equations for the horizontal asymptotes, vertical asymptotes or holes in each of the functions below. If none eist, write none. Look for patterns in the types of asymptotes that occur so you can answer the questions on the net page. X 1 X 4 1. f(x)=. f()=. f()= 4. f()= X 1 X 1 4 v.a v.a v.a v.a hole: hole: hole: hole: h.a h.a h.a h.a f()= 6. f()= 7. f()= 8. f()= 9 16 v.a v.a v.a v.a hole: hole: hole: hole: h.a h.a h.a h.a f()= 10. f()= 11. f()= 1. f()= 8 ( ) 8 v.a v.a v.a v.a hole: hole: hole: hole: h.a h.a h.a h.a

5 PACKET Unit 4 Honors ICM Functions and Limits 5 QUESTIONS: 1. Give eamples of functions that had no vertical asymptotes.. Why doesn t every function with a denominator have vertical asymptotes?. In general, how do you find vertical asymptotes algebraically? 4. List all the functions that did not have horizontal asymptotes. 5. What do the functions that have no horizontal asymptotes have in common? 6. List all the functions that had horizontal asymptotes of y = What do functions that had horizontal asymptotes of y = 0 have in common? 8. List all other functions with horizontal asymptotes that have not already been listed. 9. What do all other functions with horizontal asymptotes that have not already been listed have in common? 10. Eamine the equations of functions with and without horizontal asymptotes. What is a quick way to determine their horizontal asymptotes looking only at the equation, without the help of a graph or table? Summarize how to find the equation of a horizontal asymptote for any rational function.

6 PACKET Unit 4 Honors ICM Functions and Limits 6 Homework Day In Eercises 17 and 19, find the domain, range and end behavior. Write the end behavior using limits. 17. f ( ) f( ) 1 Domain: Domain: Range: End Behavior: Range: End Behavior: In Eercises 55-61, find all horizontal and vertical asymptotes of the function. 55. f( ) 1 HA: VA: 57. g ( ) HA: VA: 59. f( ) 1 HA: VA: g ( ) 8 HA: VA: In Eercises 6-66, match the function with the corresponding graph. 6. y 64. y 65. y y Can a graph cross its own asymptote? The Greek roots of the word asymptote mean not meeting, since graphs tend to approach, but not meet, their asymptotes. Which of the following functions have graphs that do intersect their horizontal asymptotes? (a) f( ) (b) g ( ) (c) h ( ) Eplain why a graph cannot have more than horizontal asymptotes.

7 PACKET Unit 4 Honors ICM Functions and Limits 7 Homework Day 4 In Eercises 5-8, state whether each labeled point identifies a local minimum, a local maimum, or neither (write beside each point). Identify intervals on which the function is decreasing and increasing In Eercises 9-, graph the function and identify intervals on which the function is increasing, decreasing, or constant. 9. f ( ) 1 Constant:

8 PACKET Unit 4 Honors ICM Functions and Limits 8 1. g( ) 1. h( ) ( 1) Constant: Constant: In Eercises 41-45, use a calculator to find all local maima and minima and the values of where they occur. Give values rounded to two decimal places. 41. f ( ) 4 Minima: Maima: 4. g ( ) Minima: Maima: 45. h ( ) 4 Minima: Maima:

9 PACKET Unit 4 Honors ICM Functions and Limits 9 Homework Day 6 Quiz Review Determine the following for the given function (#1-0). f( ) ) Domain: 10) vertical asymptotes: ) Range: 11) horizontal asymptotes ) removable point of 1) Continuous? discontinuity: 4) Increasing 14) lim f( ) 1) Nonremovable discontinuity? 5) Decreasing 15) lim f( ) 6) Local Min 16) 7) Local Ma 17) lim f( ) 4 lim f( ) 4 8) -intercept(s): 18) 4 lim f( ) 9) y-intercept(s): 19) lim f( ) 4 0) Sketch f() 1) Given: Find Domain (no decimals): Find Range: ) Given: Find Domain: Find Range:

10 PACKET Unit 4 Honors ICM Functions and Limits 10 Homework Day 8 In Eercises 47-5, state whether the function is odd, even, or neither. Support graphically (sketch) and confirm algebraically. 47. f ( ) g( ) 49. f 50. f( ) 1 ( ) 51. f ( ) f ( ) g( ) 7. True / False. A relation that is symmetric with respect to the -ais cannot be a function. Justify your answer. In Eercises 1-4, find formulas for the functions f g, f g, and fg. Then, give the domain of each function and the domain of each combined function. 1. f ( ) 1; g( ) FORMULAS DOMAIN f() Domain: f g: g() Domain: f g : fg :

11 PACKET Unit 4 Honors ICM Functions and Limits 11. f ( ) ( 1) ; g( ) FORMULAS DOMAIN f() Domain: f g: g() Domain: f g : fg :. f ( ) ; g( ) sin FORMULAS DOMAIN f() Domain: f g: g() Domain: f g : fg : 4. f ( ) 5; g( ) FORMULAS DOMAIN f() Domain: f g: g() Domain: f g : fg : In Eercises 5 and 6, find formulas for f / g and g / f. Give the domain of each functions and each combined function. 5. f ( ) ; g( ) FORMULAS DOMAIN f() Domain: f / g : g() Domain: g/ f :

12 PACKET Unit 4 Honors ICM Functions and Limits 1 6. f ( ) ; g( ) 4 FORMULAS DOMAIN f() Domain: f / g : g() Domain: g/ f : In Eercise 9, find ( f g)() and ( g f )( ) 9. f ( ) ; g( ) 1 a. ( f g)() b. ( g f)( ) In Eercises 11-1, find f ( g( )) and g( f ( )). Then, state the domain of each. 11. f ( ) ; g( ) 1 a. ( f ( g( )) b. ( g( f ( )) Domain: Domain: 1. f ( ) ; g( ) 1 a. ( f ( g( )) b. ( g( f ( )) Domain: Domain: In Eercises 15-19, find an epression for f ( ) and g( ) so that the function can be described as y f ( g( )) your work to verify that y f ( g( )) for the f ( ) and g( ) that you have selected.. Show 15. y y 19. y 5 ( )